let the measure of $\widehat{bcd} = a^{\circ}$. because $\widehat{bcd}$ and $\widehat{bad}$ form a circle, and a circle measures $360^{\circ}$, the measure of $\widehat{bad}$ is $360 - a^{\circ}$. because of the ______ theorem, $m\angle a = \frac{a}{2}$ degrees and $m\angle c = \frac{360 - a}{2}$ degrees. the sum of the measures of angles a and c is $\left(\frac{a}{2} + \frac{360 - a}{2}\
ight)$ degrees, which is equal to $\frac{360^{\circ}}{2}$, or $180^{\circ}$. therefore, angles a and c are supplementary because their measures add up to $180^{\circ}$. angles b and d are supplementary because the sum of the measures of the angles in a quadrilateral is $360^{\circ}$. $m\angle a + m\angle c + m\angle b + m\angle d = 360^{\circ}$, and using substitution, $180^{\circ} + m\angle b + m\angle d = 360^{\circ}$, so $m\angle b + m\angle d = 180^{\circ}$.

what is the missing information in the paragraph proof?

- inscribed angle
- polygon interior angle sum
- quadrilateral angle sum
- angle bisector

Question

let the measure of $\widehat{bcd} = a^{\circ}$. because $\widehat{bcd}$ and $\widehat{bad}$ form a circle, and a circle measures $360^{\circ}$, the measure of $\widehat{bad}$ is $360 - a^{\circ}$. because of the ______ theorem, $m\angle a = \frac{a}{2}$ degrees and $m\angle c = \frac{360 - a}{2}$ degrees. the sum of the measures of angles a and c is $\left(\frac{a}{2} + \frac{360 - a}{2}\
ight)$ degrees, which is equal to $\frac{360^{\circ}}{2}$, or $180^{\circ}$. therefore, angles a and c are supplementary because their measures add up to $180^{\circ}$. angles b and d are supplementary because the sum of the measures of the angles in a quadrilateral is $360^{\circ}$. $m\angle a + m\angle c + m\angle b + m\angle d = 360^{\circ}$, and using substitution, $180^{\circ} + m\angle b + m\angle d = 360^{\circ}$, so $m\angle b + m\angle d = 180^{\circ}$.

what is the missing information in the paragraph proof?

- inscribed angle
- polygon interior angle sum
- quadrilateral angle sum
- angle bisector

Accepted Answer

{ "quality": "clear", "question_count": 1, "skills_matched": [ "step_cot", "multiple_choice_formatter", "explain_then_answer", "quick" ], "needs_drawing": false, "concepts_used": [ "Inscribed Angle Theorem" ], "new_concepts": [], "current_concepts": [ "Inscribed Angle Theorem", "Circle Geometry" ] } ## Identify the missing theorem in the proof The proof states: "Because of the ______ theorem, $m\angle A = \frac{a}{2}$ degrees and $m\angle C = \frac{360-a}{2}$ degrees." ## Analyze the relationship between the angles and arcs - Angle $A$ is an inscribed angle that intercepts the arc $\widehat{BCD}$, which has a measure of $a^\circ$. - Angle $C$ is an inscribed angle that intercepts the arc $\widehat{BAD}$, which has a measure of $(360-a)^\circ$. - The theorem stating that the measure of an inscribed angle is half the measure of its intercepted arc is the **inscribed angle** theorem. ## Match with the given options - "inscribed angle" matches the blank.

inscribed angle

polygon interior angle sum

quadrilateral angle sum

angle bisector

{ "subject": "Mathematics", "question_type": "Multiple Choice", "knowledge_point": [ "Mathematics", "Geometry", "Inscribed Angle Theorem" ] }

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QUESTION IMAGE

Question

Explanation:

Identify the missing theorem in the proof

Analyze the relationship between the angles and arcs

Match with the given options

Answer: